Bayesian Response-Adaptive Randomisation with Group Sequential Designs
Corey Voller
April 29, 2025
Objectives
Feedback on the slides/presentation
Bayesian/Frequentist stopping boundary
Explanation of differences using different priors
Next steps
APTS Durham
APTS Durham
Use version control (Git)
APTS Durham
Attend lots of seminars
Use version control (Git)
APTS Durham
Read the latest and first paper on a topic
Attend lots of seminars
Use version control (Git)
APTS Durham
Get the simplest case working
Read the latest and first paper on a topic
Attend lots of seminars
Use version control (Git)
APTS Durham
Get the simplest case working Otherwise, go back to the drawing board
Read the latest and first paper on a topic
Attend lots of seminars
Use version control (Git)
Presentation of results
Change from plots of \(\theta\) to plots of ratios.
Include plots which overlay frequentist and priors using pooled with facet for each analysis.
Do the same for the group based approach
Stopping Boundary
Stopping Boundary
Stopping Boundary
Bayesian stopping rule
Stopping Boundary
Bayesian stopping rule
Bayesian allocation ratio only
Stopping Boundary
Bayesian stopping rule
Bayesian allocation ratio only
Stopping Boundary
Irving John Good (1916 - 2009)
The subjectivist (i.e. Bayesian) states his judgements, whereas the objectivist sweeps them under the carpet by calling assumptions knowledge, and he basks in the glorious objectivity of science.
Stopping Boundary
Irving John Good (1916 - 2009)
The subjectivist (i.e. Bayesian) states his judgements, whereas the objectivist sweeps them under the carpet by calling assumptions knowledge, and he basks in the glorious objectivity of science.
What assumptions do we want to make?
Interpretability
Option 3
Time constraint
Prior on individual treatments vs treatment effect directly
Maximum likelihood estimate
Maximum likelihood estimate
Generate data from group j using independent normal distributions with mean \(\mu_j\) and a known common variance \(\sigma^2\).
\[
X_{i,j} \sim N(\mu_j, \sigma^2)
\]
where \(j=1,2\) and \(i=1,\ldots,n_j\).
Maximum likelihood estimate
Generate data from group j using independent normal distributions with mean \(\mu_j\) and a known common variance \(\sigma^2\).
\[
X_{i,j} \sim N(\mu_j, \sigma^2)
\]
where \(j=1,2\) and \(i=1,\ldots,n_j\).
Using pooled data over groups, we get the following expression for our estimate of the treatment effect \(\theta\) at analysis \(k\)
posterior_mean <-function(prior_mean, prior_tau, data, tau, n,k) { result <-tryCatch({# Check if the lengths of n and data are compatibleif (length(n) !=length(data)) {stop("Length of n and data must be the same. Found n of length ", length(n), " and data of length ", length(data)) }# Calculate post mean posterior_mean_result <- ((prior_tau) * prior_mean +sum(n * (tau) * (data))) / (sum(n* tau) + prior_tau)return(posterior_mean_result) }, error =function(e) {# Catch the error and return a messagecat("Error in posterior_mean (K=):",k, e$message, "\n")cat("n: ",paste(n),"\n")cat("data: ",paste(data),"\n")return(NA) # Return NA or any default value in case of error })return(result)}x_1[k] <-rnorm(1, mean = mu_1, sd =sqrt(sigma ^2/ n1.new[k]))x_2[k] <-rnorm(1, mean = mu_2, sd =sqrt(sigma ^2/ n2.new[k]))# Posterior variancestau_n1[k] <-posterior_tau(prior_tau = tau_01,tau = tau,n = n1.new[1:k])tau_n2[k] <-posterior_tau(prior_tau = tau_02,tau = tau,n = n2.new[1:k])# Posterior meansmu_n1[k] <-posterior_mean(prior_mean = mu_01,n = n1.new[1:k],tau = tau,prior_tau = tau_01,data = x_1[1:k],k = k)mu_n2[k] <-posterior_mean(prior_mean = mu_02,n = n2.new[1:k],tau = tau,prior_tau = tau_02,data = x_2[1:k],k = k)theta_hat[k] <- mu_n1[k] - mu_n2[k]
Prior on individual \(\mu\)’s
No early stopping using frequentist and bayesian methods with pooled θ
EN1_pool
EN2_pool
total_pool
EN1_brar_1
EN2_brar_1
total_brar_1
EN1_brar_2
EN2_brar_2
total_brar_2
EN1_brar_3
EN2_brar_3
total_brar_3
EN1_brar_4
EN2_brar_4
total_brar_4
-1/2𝛿
86.4 (0)
122.8 (0)
209.2 (0)
86.4 (0)
122.8 (0)
209.2 (0)
94.5 (0)
107.9 (0)
202.5 (0)
111 (0)
91.3 (0)
202.4 (0)
83.3 (0)
125 (0)
208.3 (0)
0𝛿
101.4 (0)
101.5 (0)
202.9 (0)
101.5 (0)
101.4 (0)
202.9 (0)
107.8 (0)
94.7 (0)
202.5 (0)
121.1 (0)
85.5 (0)
206.6 (0)
83.3 (0)
125 (0)
208.3 (0)
1/2𝛿
122.8 (0)
86.4 (0)
209.2 (0)
122.8 (0)
86.4 (0)
209.2 (0)
126.1 (0)
84 (0)
210.1 (0)
133.8 (0)
80.1 (0)
214 (0)
83.4 (0)
125 (0)
208.3 (0)
1𝛿
153 (0)
75.7 (0)
228.7 (0)
153.1 (0)
75.7 (0)
228.7 (0)
151.5 (0)
75.5 (0)
227 (0)
150.4 (0)
75.2 (0)
225.6 (0)
83.4 (0)
124.9 (0)
208.3 (0)
3/2𝛿
195.8 (0)
68.2 (0)
263.9 (0)
195.8 (0)
68.2 (0)
264 (0)
187.7 (0)
68.8 (0)
256.5 (0)
172.6 (0)
70.6 (0)
243.2 (0)
83.4 (0)
124.9 (0)
208.3 (0)
2𝛿
256.4 (0.1)
62.8 (0)
319.3 (0.1)
256.4 (0.1)
62.8 (0)
319.2 (0.1)
239.6 (0)
63.7 (0)
303.2 (0)
203.1 (0)
66.6 (0)
269.6 (0)
83.4 (0)
124.9 (0)
208.3 (0)
Based on1e+06Simulations; mean (standard error) to 1 decimal place
The posterior variance of \(\theta\) when separate prior distributions are given is always smaller than that when only the prior distribution for \(\theta\) is used.1
Differences in using direct vs individual priors
Assess difference using idea of coupling
Primarily interested in the vague prior case
Differences in using direct vs individual priors
Distribution of theta_hat at each analysis K using individual priors for treatments and a prior for theta directly using coupling
Differences in using direct vs individual priors
Means and variances of theta_hat at each analysis K using individual priors for treatments and a prior for theta directly using coupling
Conferences
Three conference deadlines in May:
Samba Bath 2nd May 2025
RSS Rapid-fire talks Edinburgh 9th May 2025
Bayes Pharma Netherlands 15th May
Stallard, Nigel, Susan Todd, Elizabeth G. Ryan, and Simon Gates. 2020. “Comparison of Bayesian and Frequentist Group-Sequential Clinical Trial Designs.”BMC Medical Research Methodology 20 (1): 4. https://doi.org/10.1186/s12874-019-0892-8.